The PMSU Spectroscopic Survey of Nearby M dwarfs

1. Hipparcos and the upper main sequence - a volume-limited sample

The ESA Hipparcos satellite obtained milli-arcsecond accuracy astrometry for over 118,000 stars over the whole
sky. The satellite carried out a pointed survey, targeting individual objects from an input catalogue,
compiled in the early 1980s based on proposals made by the European astronoical community.
While that catalogue includes a few stars (and 3C273) fainter than V=12, the overwhelming majority
of stars are brighter than 9th magnitude. The formal completeness limit is latitude dependent

V = 7.9 + 1.1 sin|b|

but, since the input catalogue included a series of proposals which specifically targeted known and
suspected nearby stars, that sample is likely to be more complete. Jahreiss & Wielen (1997) estimate that
the catalogue is effectively complete for stars brighter than MV=8.5 within 25 partsecs of
the Sun. Consequently, the Hipparcos dataset almost exactly complements the PMSU survey, and allows us to
extend coverage to brighter, higher-mass stars in the Solar Neighbourhood.

There are a total of 1477 stars in the Hipparcos catalogue with parallaxes larger than 40 mas, consistent
with distances of less than 25 parsecs. As Figure P.8 shows, besides upper main-sequence and evolved
stars, that sample includes early- and mid-type M dwarfs and white dwarfs, together a number of stars lying
at somewhat unusual locations in the HR diagram. Most of the latter stars have parallax measurements
with large formal undertainties, usually due to the presence of a close (physical or line-of-sight)
companion. Eliminating those stars from the sample (only a couple are formally brighter than
MV=8.0) leaves 844 stars. That sample includes red giants and metal-poor subdwarfs, as well as
separate measurements for components in wide binary systems. Since our aim is a derivation of the
luminosity function for main-sequence stars and systems including main-sequence components, the sample
needs to be trimmed of those unwanted contributors.

The resulting datasets are listed at the foot of this page: there are 41 evolved stars (where
"evolved" is defined based on the location on the HR diagram, see Figure P.8); 4 subdwarfs (HD 25329,
HD 103095, HD 120559 and HD 145417, with -0.9 < [Fe/H] < -1.6); and measurements of 35 secondary
components. We have also searched the literature to identify binary/multiple systems and determine,
as far as possible, their apparent and absolute magnitudes.
Figure P.8 plots the (MV, (B-V)) colour-magnitude diagram outlined by the
remaining 764 primaries or single stars. Where necessary, the magnitudes listed by Hipparcos have been
adjusted to allow for the contribution of unresolved secondaries, so a few stars slipe below
the nominal MV=8.0 limit. As discussed below, several of those stars are not in the PMSU
sample.

Having trimmed the sample to main-sequence stars only, we can test Jahreiss & Wielen's conjecture
that the Hipparcos catalogue is complete to 25 parsecs for stars with MV < 8.5. Figure P.9
plots the average density as a function of distance for systems including main-sequence stars with absolute magnitudes
in the range 4 < MV < 8. The innermost point plots the average density for d < 16 pc.; subsequent
points plot the density in spherical shells from 16 to 18, 18 to 20, 20 to 22 and 22 to 25 parsecs. There is
no evidence for a significant decline in densities in the outermost shell, even for the faintest stars.

Figure P.9: The run of number density with distance
for Hipparcos main-sequence stars with 4 < MV < 8.

2. Multiplicity

The 25-parsec upper main-sequence sample includes 764 systems: 538 single stars, 204 binaries
(including 8 low-amplitude spectroscopic binaries from the Keck/Lick survey, summarised by
Nidever sl et al., 2002), 22 triples and 4 quadruple systems. The resultant multiplicity
fraction is only 30.1+/-2.4 %, somewhat lower than the 44 % derived by Duquennoy & Mayor (1991; DM91)
in what has become the standard reference for multiplicity amongst solar-type stars. This could
reflect incompleteness in surveying the Hipparcos sample - and there are at least 14 % of
the sample which lack any radial velocity measurements - but it might also reflect the fact that
the sample extends to fainter, lower-mass stars than the DM91 survey.
M dwarfs have a lower multiplicity, between 30 and 35% - see Fischer & Marcy, 1992; Reid & Gizis, 1997).
Finally, there is a possibility that the DM91 sample could be biased. The intention was to
pick a volume-limited sample, but since the stars were selected from the CNS2, the parallax
measurements were subject to the systematics illustrated in
Figure P.5 . Since all of the stars were observed by Hipparcos, we can check whether such
a bias exists.

Figure P.10: The DM91 sample. The lower panel
compares the CNS2 and Hipparcos parallax measurements, with the nominal distance limit of 45 mas outlined.
The upper panel plots the distribution in the HR diagram. Magenta points are single stars, green points
are known binaries.

The DM91 sample includes stars with spectral types between F7 and G9, parallaxes exceeding 45 mas
and declinations above -15o. Figure P.10 compares the CNS2 and Hipparcos parallax measurements
for the 169 systems from DM91 which meet those criteria. Only 102 actually lie within the formal
distance limit. However, 42 of those 102 systems are double or multiple, so the multiplicity fraction
is 41+/-7 %, statistically identical with DM91's calculation (and only 1.5 sigma above the
Hipparcos 25-parsec binary fraction). Thus observational bias is not a likely
source of the difference between the DM91 and Hipparcos datasets.

Given this consistency check, we can use DM91's analysis of their sample to estimate a likely
upper limit to the binary fraction in the Hipparcos sample. Low-amplitude spectroscopic binaries,
either wide, long-period systems, high inclination systems, or high mass-ratio systems,
are more difficult to detect. Given the available statistics, DM91 calculated that their observations
underestimated the true binary fraction by about one third, deriving a total multiplicity of 57 %.
In passing, we note that 47 of the 60 "single" stars from the revised DM91 sample are on
the Keck/Lick program; only one has a velocity r.m.s. exceeding 100 m/sec. Nonetheless, this
value provides a reasonable upper limit to the likely contribution of binaries to the Hipparcos
sample - so we have allowed for that possibility by giving double weight to each of the known
binary companions.

3. The luminosity function

Our goal is constructing a luminosity function - the number of stars/systems per unit
absolute magnitude per unit volume. The statistics are listed in the following table, combining
both the Hipparcos 25-parsec and PMSU datasets. The distance limits for companions to Hipparcos 25-parsecs
primaries are based onMV (companion), and match those adopted for the PMSU sample.

Table P4: The luminosity function for nearby stars

MV

Hipparcos

PMSU4

Phisys

Phistars

Nsys

Nsec(1)

Nsec(2)

Nsys

Nsec(1)

Nsec(2)

Nsupp

systems pc-3 x 104

stars pc-3 x 104

-0.5

3

0.46

0.46

0.5

5

1

1

0.76

0.92

1.5

15

2

2

2.29

2.60

2.5

36

1

1

5.50

5.65

3.5

79

9

9

12.07

13.45

4.5

151

15

15

23.07

25.36

5.5

147

19

19

22.46

25.36

6.5

181

32

32

27.65

32.54

7.5

143

24

24

21.85

25.52

8.5

8

41

27

103

7

7

6

32.01

40.16

9.5

23

13

92

7

5

3

50.21

58.26

10.5

28

4

64

15

9

74.24

88.16

11.5

21

2

66

17

11

3

76.56

91.06

12.5

15

6

71

25

17

1

82.36

107.30

13.5

7

23

14

6

73.21

92.31

14.5

13

6

3

41.38

50.93

15.5

5

1

4

12

5

101.86

248.28

16.5

1

2

2

25.46

76.39

17.5

2

6

1

50.93

76.39

Column 2 lists the number-magnitude distribution
of single stars and primary stars in the Hipparcos 25-parsec sample;
column 3 gives the magnitude distribution for known companions; column
4 lists the companion star distribution after applying the
absolute magnitude-dependent distance limits given in Table 1. Columns
5, 6 and 7 provide the same statistics for the PMSU4 sample, and column
8 gives the contribution from the supplementary stars listed in
Table 3. Combining
the samples, the luminosity function due to primaries and single stars
is listed in column 9, while the space densities given in column 10
include the contribution from the companions listed in columns 4
and 7.

Figure P.11 compares the resulting luminosity function against the results derived in 1983 by
Wielen et al, using data from the CNS2 and its supplement. The current luminosity function tends to
lie below the older analysis, reflecting the systematic errors in pre-Hipparcos parallax measurements,
which tended to place stars nearer the Sun, and therefore inflated the local density estimates. Those
effects are particularly evident amongst the M dwarfs - as discussed previously. Figure P.12 shows the
effect of giving double weight to the Hipparcos binaries; the resulting function lies
somewhat closer to the Wielen et al calculation at brighter magnitudes, but the overall effect is not
particularly significant.

Figure P.12: The nearby-star luminosity function derived
when double weight is assigned to the Hipparcos 25-parsec secondaries. The symbols have the same meaning
as in Figure P.11. The effect is not particularly significant.

4. The mass-luminosity relation

Given a main-sequence luminosity function, we need a mass-luminosity relation to transform the
results to give a mass function. That relation is defined using observations of stars in eclipsing
or astrometric binaries. In general, eclipsing binaries provide the calibrating stars at high masses
(> 1 MSun); astrometric binaries, plus a handful of eclipsing systems (YY Gem, CM Dra,
GJ 2069) provide the calibration at sub-solar masses.
Henry & McCarthy (1993) undertook the first thorough analysis
extending to the hydrogen-burning limit, matching the available data with a three-component fit.
Since that study, data have become available for more binary systems, while continued
observations have led to better orbit determinations, and better mass estimates, for other sysyems
(Segresan et al, 2001). Those improved data have been analysed by Delfosse et al (2001) to
derive revised (mass, absolute magnitude) relations. Figure P.13 compares their (mass, MV) relation
against the Henry & McCarthy dataset, and also shows a polynomial fit to Andersen's compilation of data for
higher-mass systems.

Figure P.13: The mass-luminosity relation: cyan point mark
data for eclipsing binaries from Andersen (1991); green points plot the revised masses for low-mass
systems from the compilation by segresan et al (2001); the dotted line marks Delfosse et al's (2001) fit
to the low-mass data; the dashed line is a fit to the Andersen dataset; and the solid line plots the
Henry & McCarthy relation.

Kroupa, Tout & Gilmore (1993) followed a different approach to deriving a (mass, MV)
relation. Rather than fit the data directly, they adopt a reference luminosity function, and
choose a functional form for the mass function (a three-component power-law, with breaks at
0.5 and 1.0 solar masses). The (mass, MV) relation is then varied to give the
lowest residuals with respect to the observed datapoints. As Figure P.14 shows,
The resulting relation is a close match to
the Delfosse et al empirical fit at masses below ~0.5 MSun, but predicts lower masses
than all fo the other relations for 8 > MV > 3. It does provide a better match to the
fiducial solar point.

Figure P.14: The mass-luminosity relation: the symbols have the
same meaning as in Figure P.13, withe the addition of the blue long-dashed line marking the KTG
(mass, MV) relation, and the red star marking the location of the Sun.

5. The present-day mass function

Figure P.15 shows the results of applying (mass, MV) relations to the nearby-star data.
In each case, masses are derived on a star by star basis - i.e. we use the (mass, MV) relation
to estimate a mass for each star, then combine the results to deribe the mass function.
There are two composite (mass, MV) relations: the Delfosse et al fit combined with the
empirical fit to the Andersen dataset, with the break between the two set at MV = 10.0; and
the KTG relation combined with the Andersen fit, with the break set at MV=3.5. The former
calibration is used to analyse both the straight observational dataset (i.e. the data used to derive
Figure P.11) and the dataset where double weight is given to the Hipparcos 25-parsec secondaries
(the data used to derive Figure P.12).

Figure P.15: The present-day mass function: the uppermost
panel plots the results of transforming the luminosity function plotted in Figure P.11 to masses
using a composite (mass, MV) relation derived by combining the Delfosse et al fit with
our polynomial fit to the Andersen dataset; the middle panel shows the effect of giving double
weight to the Hipparcos binaries (i.e. applying the same conversion to the Figure P.12 data); the
lowermost panel shows the mass function derived by combining the KTG and Andersen fits,

The resulting mass functions are plotted in Figure P.16, and are in generally good agreement.
Fitting the results with power-law mass functions, where alpha=2.35 is the Salpeter
slope, we find:

It is not surprising that the two changes of slope in the last mass function match those used to
define the (mass, MV) relation.

6. The initial mass function

The mass functions derived in the previous section reflect the distribution of masses amongst
stars in the Solar Neighbourhood at the present time. A more fundamental parameter is the
initial mass function - the distribution of masses at the time of formation. We can derive that
function from the present-day mass function by applying three corrections:

a correction for the variation in main-sequence lifetime with stellar mass - higher-mass
stars have lifetimes shorter than the age of the galactic disk, so only the most recently formed
stars remain on the main sequence. In order to compensate for the evolved fracton, we need not
only estimates of the main-sequence lifetimes (which can be derived from evolutionary models), but
also an estimate of the star formation history of the disk. We assume a constant star formation rate
and take the main-sequence lifetimes from Schaller et al (1992).

a correction for the distribution of stars perpendicular to the Galactic Plane - velocity
dispersion increases with age, and the vertical scaleheight varioes accordingly. We have modelled
the distribution as an exponential, scaleheight 250 parsecs for MV > 4; 170
parsecs for 3 < MV < 4; and 100 parsecs for the brightest stars (see Siegel et al, 2002).
For this density distribution, the local surface density is simply the volume density multiplied
by twice the scaleheight.

a correction for the local mix of populations - about 10% of local stars are members of the
thick disk population, an old stellar population with a more extended vertical density distribution.
We allow for their contribution to the local numberdensity of stars with MV > 4.

Figure P16 shows the IMFs derived from the Andersen+Delfosse et al and Andersen+KTG present-day
mass functions. The two functions are very similar in form, both well-matched by two power-law
components, with the main difference being the location of the change in slope. The parameters are
as follows:

In both cases, thick disk dwarfs contribute a further 15 solar masses / pc2 (placing 10% of local
stars in the thick disk, with a scaleheight of 1 kiloparsec).

Neither of the IMFs show evidence for a change in the power-law index
near the hydrogen-burning limit. As discussed in more detail by Reid {\sl et al.} (1999),
direct determination of the mass function at sub-stellar masses
in the field is complicated by the short visible lifetimes of brown dwarfs, and the
consequent necessity of extrapolating the observed numbers to take account of systems which
have cooled below the detection limit. In principle, this is the same issue affecting
upper-main sequence stars; in practice, the uncertainties are more severe, since all brown
dwarfs effectively follow the same cooling track in the HR diagram. As a result, it is
not possible to assign a mass to an individual brown dwarf without prior knowledge of its
age. Moreover, the
the slower cooling time of higher-mass brown dwarfs means that they effectively dominate in
any magnitude-limited sample. The current best estimate for the field, based on
comparing simulations to the observed surface density of L and T dwarfs in the
2MASS and SDSS surveys, gives alpha ~ 1.3; more significantly, alpha < 2, so there is
no question of hiding substantial amounts of dark matter as disk brown dwarfs (Reid et al, 1999).
Observations of young clusters suggest a somewhat flatter distribution, 0.5 < alpha < 1.

The Hipparcos 25-parsec sample

All files are ascii, and most include the following parameters:
Hipparcos number, MV. VT, BT, (B-V)T,sigmaV,
sigmaB, Vlit, (B-V)lit, pi, sigmapi, muRA,
mudec, sigmamu(RA), sigmamu(Dec), RA, Dec
MV is based on the literature photometry, and the astrometric parameters are
in milliarcseconds. Some of the files also list a more
common identifier for individual stars. The list of companions gives
Hipparcos number of primary, V(companion), MV, distance (pc), binary type, spectral
type, name and comments
Some of the apparent (hence absolute) magnitudes and spectral types are estimated rather than
observed.